Stability of Nonlinear Systems
Stability of Nonlinear Systems
By
Guanrong Chen
Department of Electronic Engineering City University of Hong Kong Kowloon, Hong Kong SAR, China [email protected]
In
Encyclopedia of RF and Microwave Engineering, Wiley, New York, pp. 4881-4896, December 2004
S Continued
STABILITY OF NONLINEAR SYSTEMS librium position of the system is stable [23]. The evolution of the fundamental concepts of system and trajectory sta- bilities then went through a long history, with many fruit- GUANRONG CHEN City University of Hong Kong ful advances and developments, until the celebrated Ph.D. Kowloon, Hong Kong, China thesis of A. M. Lyapunov, The General Problem of Motion Stability, finished in 1892 [21]. This monograph is so fun- damental that its ideas and techniques are virtually lead- 1. INTRODUCTION ing all kinds of basic research and applications regarding stabilities of dynamical systems today. In fact, not only A nonlinear system refers to a set of nonlinear equations dynamical behavior analysis in modern physics but also (algebraic, difference, differential, integral, functional, or controllers design in engineering systems depend on the abstract operator equations, or a combination of some of principles of Lyapunov’s stability theory. This article is these) used to describe a physical device or process that devoted to a brief description of the basic stability theory, otherwise cannot be clearly defined by a set of linear equa- criteria, and methodologies of Lyapunov, as well as a few tions of any kind. Dynamical system is used as a synonym related important stability concepts, for nonlinear dynam- for mathematical or physical system when the describing ical systems. equations represent evolution of a solution with time and, sometimes, with control inputs and/or other varying pa- 2. NONLINEAR SYSTEM PRELIMINARIES rameters as well. The theory of nonlinear dynamical systems, or nonlin- 2.1. Nonlinear Control Systems ear control systems if control inputs are involved, has been greatly advanced since the nineteenth century. Today, A continuous-time nonlinear control system is generally nonlinear control systems are used to describe a great va- described by a differential equation of the form riety of scientific and engineering phenomena ranging . from social, life, and physical sciences to engineering x ¼ fðx; t; uÞ; t 2½t0; 1Þ ð1Þ and technology. This theory has been applied to a broad spectrum of problems in physics, chemistry, mathematics, where x ¼ x(t) is the state of the system belonging to a n biology, medicine, economics, and various engineering dis- (usually bounded) region Ox R , u is the control input ciplines. vector belonging to another (usually bounded) region Ou Stability theory plays a central role in system engineer- Rm (often, m n), and f is a Lipschitz or continuously dif- ing, especially in the field of control systems and automa- ferentiable nonlinear function, so that the system has a tion, with regard to both dynamics and control. Stability of unique solution for each admissible control input and suit- a dynamical system, with or without control and distur- able initial condition x(t0) ¼ x0AOx. To indicate the time bance inputs, is a fundamental requirement for its prac- evolution and the dependence on the initial state x0, the tical value, particularly in most real-world applications. trajectory (or orbit) of a system state x(t) is sometimes Roughly speaking, stability means that the system out- denoted as jt(x0). puts and its internal signals are bounded within admissi- In control system (1), the initial time used is t0 0, ble limits (the so-called bounded-input/bounded-output unless otherwise indicated. The entire space Rn, to which stability) or, sometimes more strictly, the system outputs the system states belong, is called the state space. Associ- tend to an equilibrium state of interest (the so-called as- ated with the control system (1), there usually is an ob- ymptotic stability). Conceptually, there are different kinds servation or measurement equation of stabilities, among which three basic notions are the main concerns in nonlinear dynamics and control systems: y ¼ gðx; t; uÞð2Þ the stability of a system with respect to its equilibria, the orbital stability of a system output trajectory, and the where y ¼ yðtÞ2R‘ is the output of the system, 1 ‘ n, structural stability of a system itself. and g is a continuous or smooth nonlinear function. When The basic concept of stability emerged from the study of both n, ‘>1, the system is called a multiinput/multioutput an equilibrium state of a mechanical system, dated back to (MIMO) system; while if n ¼ ‘ ¼ 1, it is called a single- as early as 1644, when E. Torricelli studied the equilibri- input/single-output (SISO) system. MISO and SIMO um of a rigid body under the natural force of gravity. The systems are similarly defined. classical stability theorem of G. Lagrange, formulated in In the discrete-time setting, a nonlinear control system 1788, is perhaps the best known result about stability of is described by a difference equation of the form conservative mechanical systems, which states that if the ( potential energy of a conservative system, currently at the xk þ 1 ¼ fðxk; k; ukÞ position of an isolated equilibrium and perhaps subject to k ¼ 0; 1; ð3Þ some simple constraints, has a minimum, then this equi- yk ¼ gðxk; k; ukÞ; 4881 4882 STABILITY OF NONLINEAR SYSTEMS where all notations are similarly defined. This article usu- at x~ ðtÞ. Then this Jacobian is also p-periodic: ally discusses only the control system (1), or the first equation of (3). In this case, the system state x is also Jðx~ ðt þ pÞÞ ¼ Jðx~ ðtÞÞ for all t 2½t0; 1Þ considered as the system output for simplicity. A special case of system (1), with or without control, is In this case, there always exist a p-periodic nonsingular said to be autonomous if the time variable t does not ap- matrix M(t) and a constant matrix Q such that the fun- pear separately (independently) from the state vector in damental solution matrix associated with the Jacobian the system function f. For example, with a state feedback Jðx~ ðtÞÞ is given by [4] control u(t) ¼ h(x(t)), this often is the situation. In this case, the system is usually written as FðtÞ¼MðtÞetQ
. n x ¼ fðxÞ; xðt0Þ¼x0 2 R ð4Þ Here, the fundamental matrix F(t) consists of, as its col- umns, n linearly independent solution vectors of the linear . Otherwise, as (1) stands, the system is said to be nonau- equation x ¼ Jðx~ ðtÞÞx, with x(t0) ¼ x0. tonomous. The same terminology may be applied in the In the preceding, the eigenvalues of the constant ma- same way to discrete-time systems, although they may trix epQ are called the Floquet multipliers of the Jacobian. have different characteristics. The p-periodic solution x~ ðtÞ is called a hyperbolic periodic An equilibrium,orfixed point, of system (4), if it exists, orbit of the system if all its corresponding Floquet multi- is a solution, x , of the algebraic equation pliers have nonzero real parts. Next, let D be a neighborhood of an equilibrium, x ,of fðx Þ¼0 ð5Þ the autonomous system (4). A local stable and local unstable manifold of x is defined by . It then follows from (4) and (5) that x ¼ 0, which means that an equilibrium of a system must be a constant state. s Wlocðx Þ¼fx 2DjjtðxÞ2D8t t0 and For the discrete-time case, an equilibrium of system ð9Þ jtðxÞ!x as t !1g xk þ 1 ¼ fðxkÞ; k ¼ 0; 1; ... ð6Þ and is a solution, if it exists, of equation u Wlocðx Þ¼fx 2DjjtðxÞ2D8t t0 and ¼ ð ÞðÞ ð10Þ x f x 7 jtðxÞ!x as t ! 1g An equilibrium is stable if some nearby trajectories of the system states, starting from various initial states, ap- respectively. Furthermore, a stable and unstable manifold proach it; it is unstable if some nearby trajectories move of x is defined by away from it. The concept of system stability with respect s s O to an equilibrium will be precisely introduced in Section 3. W ðx Þ¼fx 2DjjtðxÞ\Wlocðx Þ fgð11Þ A control system is deterministic if there is a unique consequence to every change of the system parameters or and initial states. It is random or stochastic, if there is more u u O than one possible consequence for a change in its para- W ðx Þ¼fx 2DjjtðxÞ\Wlocðx Þ fgð12Þ meters or initial states according to some probability distribution [6]. This article deals only with deterministic respectively, where f denotes the empty set. For example, systems. the autonomous system ( . 2.2. Hyperbolic Equilibria and Their Manifolds x ¼ y . Consider the autonomous system (4). The Jacobian of this y ¼ xð1 x2Þ system is defined by has a hyperbolic equilibrium (x , y ) ¼ (0, 0). The local sta- @f ble and unstable manifolds of this equilibrium are illus- JðxÞ¼ ð8Þ @x trated by Fig. 1a, and the corresponding stable and unstable manifolds are visualized by Fig. 1b. Clearly, this is a matrix-valued function of time. If the Ja- A hyperbolic equilibrium only has stable and/or unsta- cobian is evaluated at a constant state, say, x or x0, then it ble manifolds since its associated Jacobian has only stable becomes a constant matrix determined by f and x or x0. and/or unstable eigenvalues. The dynamics of an autono- An equilibrium x of system (4) is said to be hyperbolic mous system in a neighborhood of a hyperbolic equilibri- if all eigenvalues of the system Jacobian, evaluated at this um is quite simple—it has either stable (convergent) or equilibrium, have nonzero real parts. unstable (divergent) properties. Therefore, complex dy- For a p-periodic solution of system (4), x~ ðtÞ, with a fun- namical behaviors such as chaos are seldom associated damental period p40, let Jðx~ ðtÞÞ be its Jacobian evaluated with isolated hyperbolic equilibria or isolated hyperbolic STABILITY OF NONLINEAR SYSTEMS 4883
y y
s u u y W (0,0) = W (0,0) S u W loc(0,0) x x s W loc(0,0) Figure 2. The block diagram of an open-loop system.
(a)(b) measure zero). For a piecewise continuous function Figure 1. Stable and unstable manifolds. f(t), actually
ess sup jf ðtÞj ¼ sup jf ðtÞj periodic orbits [7,11,15] (see Theorem 16, below); they a t b a t b generally are confined within the center manifold, 2. The main difference between the ‘‘sup’’ and the c s c u W (x ), where dim(W ) þ dim(W ) þ dim(W ) ¼ n. ‘‘max’’ is that max|f(t)| is attainable but sup|f(t)| may not. For example, max0 to1 j sinðtÞj ¼ 1 but 2.3. Open-Loop and Closed-Loop Systems t sup0 to1 j1 e j¼1. Let S be an MIMO system, which can be linear or nonlin- 3. The difference between the Euclidean norm and the ear, continuous-time or discrete-time, deterministic or sto- Lp norms is that the former is a function of time but chastic, or any well-defined input–output map. Let U and the latters are all constants. Y be the sets (sometimes, spaces) of the admissible input 4. For a finite-dimensional vector x(t), with no N, all and corresponding output signals, respectively, both de- the Lp norms are equivalent in the sense that for any fined on the time domain D¼½a; b ; 1 aob 1 (for con- p, qA[1, N], there exist two positive constants a and N trol systems, usually a ¼ t0 ¼ 0 and b ¼ ). This simple b such that relation is described by an open-loop map
ajjxjjp jjxjjq bjjxjjp S : u ! y or yðtÞ¼SðuðtÞÞ ð13Þ For the input–output map in Eq. (13), the so-called oper- and its block diagram is shown in Fig. 2. Actually, every ator norm of the map S is defined to be the maximum gain control system described by a differential or difference from all possible inputs over the domain of the map to equation can be viewed as a map in this form. But, in such their corresponding outputs. More precisely, the operator a situation, the map S can be implicitly defined only via norm of the map S in (13) is defined by the equation and initial conditions. In the control system (1) or (3), if the control inputs are jjy1 y2jjY functions of the state vectors, u ¼ h(x; t), then the control jjjSjjj ¼ sup ð14Þ u1;u22U jju1 u2jjU system can be implemented via a closed-loop configura- u1Ou2 tion. A typical closed-loop system is shown in Fig. 3, where usually S1 is the plant (described by f) and S2 is the con- where yi ¼ S(ui)AY, i ¼ 1, 2, and || ||U and || ||Y are troller (described by h); yet they can be reversed. the norms of the functions defined on the input–output sets (or spaces) U and Y, respectively. 2.4. Norms of Functions and Operators This article deals only with finite-dimensional systems. 3. LYAPUNOV, ORBITAL, AND STRUCTURAL STABILITIES For an n-dimensional vector-valued functions, T xðtÞ¼½x1ðtÞ ... xnðtÞ where superscript ‘‘T’’ denotes trans- Three different types of stabilities, namely, the Lyapunov pose, let || || and || ||p denote its Euclidean norm stability of a system with respect to its equilibria, the or- and Lp norm, defined respectively by the ‘‘length’’ bital stability of a system output trajectory, and the struc- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tural stability of a system itself, are of fundamental 2 2 importance in the studies of nonlinear dynamics and con- jjxðtÞjj ¼ x1ðtÞþ þxnðtÞ trol systems. and Z b 1=p p jjxjjp ¼ jjxðtÞjj dt ; 1 po1 u1 + e1 S1(e 1) a S _ 1
jjxjj1 ¼ ess sup jxiðtÞj a t b 1 i n
e + Here, a few remarks are in order: S2(e 2) 2 + S2 u2 1. The term ‘‘ess sup’’ means ‘‘essential suprimum’’ (i.e., the suprimum except perhaps over a set of Figure 3. A typical closed-loop control system. 4884 STABILITY OF NONLINEAR SYSTEMS
Roughly speaking, the Lyapunov stability of a system It has an explicit solution with respect to its equilibrium of interest is about the be- havior of the system outputs toward the equilibrium 1 t0 xðtÞ¼x0 ; 0 t0 to1 state—wandering nearby and around the equilibrium 1 t (stability in the sense of Lyapunov) or gradually approach- es it (asymptotic stability); the orbital stability of a system which is stable in the sense of Lyapunov about the equi- output is the resistance of the trajectory to small pertur- librium x ¼ 0 over the entire time domain [0,N) if and bations; the structural stability of a system is the resis- only if t0 ¼ 1. This shows that the initial time, t0, does play tance of the system structure against small perturbations an important role in the stability of a nonautonomous [3,13,16,17,19,20,23,25,26,29,35]. These three basic types system. of stabilities are introduced in this section, for dynamical The above-defined stability, in the sense of Lyapunov, is systems without explicitly involving control inputs. said to be uniform with respect to the initial time, if the Consider the general nonautonomous system existing constant d ¼ d(e) is indeed independent of t0 over the entire time interval [0, N). According to the discussion . n x ¼ fðx; tÞ; xðt0Þ¼x0 2 R ð15Þ above, uniform stability is defined only for nonautono- mous systems since it is not needed for autonomous sys- where the control input u(t) ¼ h(x(t), t), if it exists [see tems (for which it is always uniform with respect to the system (1)], has been combined into the system function f initial time). for simplicity of discussion. Without loss of generality, as- sume that the origin x ¼ 0 is the system equilibrium of 3.2. Asymptotic and Exponential Stabilities interest. Lyapunov stability theory concerns various sta- bilities of the system orbits with respect to this equilibri- System (15) is said to be asymptotically stable about its um. When another equilibrium is discussed, the new equilibrium x ¼ 0, if it is stable in the sense of Lyapunov equilibrium is first shifted to zero by a change of vari- and, furthermore, there exists a constant, d ¼ d(t0)40, ables, and then the transformed system is studied in the such that same way. jjxðt0Þjjod )jjxðtÞjj ! 0ast !1 ð17Þ 3.1. Stability in the Sense of Lyapunov This stability is visualized by Fig. 5. System (15) is said to be stable in the sense of Lyapunov The asymptotic stability is said to be uniform if the ex- with respect to the equilibrium x ¼ 0, if for any e > 0 isting constant d is independent of t over [0, N), and is and any initial time t 0, there exists a constant, d ¼ 0 0 said to be global if the convergence, ||x||-0, is inde- dðe; t0Þ > 0, such that pendent of the initial state x(t0) over the entire spatial domain on which the system is defined (e.g., when d ¼ N). jjxðt0Þjjod )jjxðtÞjjoe for all t t0 ð16Þ If, furthermore This stability is illustrated by Fig. 4. st It should be emphasized that the constant d generally jjxðt0Þjjod )jjxðtÞjj ce ð18Þ depends on both e and t0. It is particularly important to point out that, unlike autonomous systems, one cannot for two positive constants c and s, then the equilibrium is simply fix the initial time t0 ¼ 0 for a nonautonomous sys- said to be exponentially stable. The exponential stability is tem in a general discussion of its stability. For example, visualized by Fig. 6. consider the following linear time-varying system with a Clearly, exponential stability implies asymptotic stabil- discontinuous coefficient: ity, and asymptotic stability implies the stability in the sense of Lyapunov, but the reverse need not be true. For . 1 illustration, if a system has output trajectory x1(t) ¼ xðtÞ¼ xðtÞ; xðt0Þ¼x0 1 t x0 sin(t), then it is stable in the sense of Lyapunov about 0, but is not asymptotically stable; a system with output
x (t 0) x(t) x (t 0) t x(t) t
Figure 4. Geometric meaning of stability in the sense of Lya- punov. Figure 5. Geometric meaning of the asymptotic stability. STABILITY OF NONLINEAR SYSTEMS 4885
A more precise concept of orbital stability is given in the sense of Zhukovskij [37].
x(t 0) A solution jt(x0) of system (19) is said to be stable in the x(t) sense of Zhukovskij, if for any e > 0, there exists a d ¼ t d(e)40 such that for any y AB (x ), a ball of radius d cen- 0 d 0 tered at x0, there exist two functions, t1 ¼ t1(t) and t2 ¼ t2(t), satisfying
jj ð Þ ð Þjj Figure 6. Geometric meaning of the exponential stability. xt1 x0 xt2 x0 oe
1 trajectory x2(t) ¼ x0(1 þ t t0) is asymptotically stable for all t t0, where t1 and t2 are homeomorphisms (i.e., a (so also is stable in the sense of Lyapunov) if t0o1 but is continuous map whose inverse exists and is also continu- not exponentially stable about 0; however, a system with N N t ous) from [0, ) to [0, ) with t1(0) ¼ t2(0) ¼ 0. x3(t) ¼ x0e is exponentially stable (hence, is both asymp- Furthermore, a Zhukovskij stable solution jt(x0) of sys- totically stable and stable in the sense of Lyapunov). tem (19) is said to be asymptotically stable in the sense of Zhukovskij, if for any e40, there exists a d ¼ d(e)40 such 3.3. Orbital Stability that for any y0ABd(x0), a ball of radius d centered at x0, The orbital stability differs from the Lyapunov stabilities there exist two functions, t1 ¼ t1(t) and t2 ¼ t2(t), satisfying in that it concerns with the stability of a system output (or state) trajectory under small external perturbations. jjxt1 ðx0Þ xt2 ðx0Þjj ! 0 Let jt(x0)beap-periodic solution, p40, of the auton- omous system
as t-N, where t1 and t2 are homeomorphisms from . n xðtÞ¼fðxÞ; xðt0Þ¼x0 2 R ð19Þ [0, N) to [0, N) with t1(0) ¼ t2(0) ¼ 0. It can be verified that (asymptotic) Zhukovskij stability and let G represent the closed orbit of jt(x0) in the state implies (asymptotic) Lyapunov stability about an equilib- space, namely, rium. However, the converse may not be true. Moreover, these two types of stabilities are equivalent if the orbit G ¼fy j y ¼ jtðx0Þ; 0 topg jt(x0) is an equilibrium of the system.
If, for any e40, there exits a constant d ¼ d(e)40 such that for any x0 satisfying 3.4. Structural Stability Two systems are said to be topologically orbitally equiva- dðx0; GÞ : ¼ inf jjx0 yjjod y2G lent, if there exists a homeomorphism that transforms the family of trajectories of the first system to that of the sec- the solution of the system, jt(x0), satisfies ond while preserving their motion directions. Roughly, this means that the geometrical pictures of the orbit fam- ilies of the two systems are similar (no one has extra dðjtðx0Þ; GÞoe; for all t t0 knots, sharp corners, bifurcating branches, etc.). For in- . . stance, systems x ¼ x and x ¼ 2x are topologically orbitally then this p-periodic solution trajectory, jt(x0), is said to be . . pffiffiffi orbitally stable. equivalent, but are not so between x ¼ x and x ¼ x. These Orbital stability is visualized by Fig. 7. For a simple three system trajectories are shown in Fig. 8. example, a stable periodic solution, particularly a stable Return to the autonomous system (19). If the dynamics equilibrium of a system is orbitally stable. This is because of the system in the state space changes radically, for ex- all nearby trajectories approach it and, as such, it becomes ample by the appearance of a new equilibrium or a new a nearby orbit after a small perturbation and so will move periodic orbit, due to small external perturbations, then back to its original position (or stay nearby). On the con- the system is considered to be structurally unstable. trary, unstable and semistable (saddle-type of) periodic orbits are orbitally unstable. x x x
Γ d ( t (x0), ) < 2 2 1 t t t Γ 0 d (x 0, ) < 0 0 (a)(b)(c) . Figure 8. Trajectories of three systems for comparison: (a) x ¼ x; . . pffiffiffi Figure 7. Geometric meaning of the orbital stability. (b) x ¼ 2x; (c) x ¼ x. 4886 STABILITY OF NONLINEAR SYSTEMS
To be more precise, consider the following set of func- boundaries of such local stability regions when this and tions: the following Lyapunov methods are applied. Moreover, it is important to note that this theorem cannot be applied to a general nonautonomous system, @gðxÞ n S¼ gðxÞjjgðxÞjjo1; o1 for all x 2 R @x since for general nonautonomous systems this theorem is neither necessary nor sufficient [36]. A simple counterex- ample is the following linear time-varying system [17,34]: If, for any g 2S, there exists an e > 0 such that the orbits of the two systems "# 2 . 1 þ 1:5 cos ðtÞ 1 1:5 sinðtÞ cosðtÞ . . xðtÞ¼ xðtÞ x ¼ fðxÞ and x ¼ fðxÞþegðxÞ 1 1:5 sinðtÞ cosðtÞ 1 þ 1:5 sin2ðtÞ pffiffiffi are topologically orbitally equivalent, then the autono- This system has eigenvalues l1;2 ¼ 0:25 j0:25 7, both mous system (19), namely, the first (unperturbed) system having negative real parts and being independent of the above, is said to be structurally stable. time variable t. If this theorem is used to judge the system, . . For example, x ¼ x is structurally stable but x ¼ x2 is the conclusion would be that the system is asymptotically not, in a neighborhood of the origin. This is because when stable about its equilibrium 0. However, the solution of the second system is slightly perturbed, to become, say, this system is . 2 x ¼ x þ e, wherepeffiffi > 0, then thep resultingffiffi system has two "#"# ¼ ¼ 0:5t t equilibria, x1 e and x2 e, which has more num- e cosðtÞ e sinðtÞ x1ðt0Þ bers of equilibria than the original system that possesses xðtÞ¼ 0:5t t only one, x ¼ 0. e sinðtÞ e cosðtÞ x2ðt0Þ
which is unstable, for any initial conditions with a bound- 4. VARIOUS STABILITY THEOREMS ed and nonzero value of x1(t0), no matter how small this initial value is. This example shows that by using the Consider the general nonautonomous system Lyapunov first method alone to determine the stability of a general time-varying system, the conclusion can be wrong. . n x ¼ fðx; tÞ; xðt0Þ¼x0 2 R ð20Þ This type of counterexamples can be easily found [35]. On the one hand, this demonstrates the necessity of other where f : D ½0;1Þ ! Rn is continuously differentiable in general criteria for asymptotic stability of nonautonomous a neighborhood of the origin, D Rn, with a given initial systems. On the other hand, however, a word of caution is state x0 2D. Again, without loss of generality, assume that this types of counterexamples do not completely rule that x ¼ 0 is a system equilibrium of interest. out the possibility of applying the first method of Lyapu- nov to some special nonautonomous systems in case stud- ies. The reason is that there is no theorem saying that ‘‘the 4.1. Lyapunov Stability Theorems Lyapunov first method cannot be applied to all nonauton- First, for the autonomous system (19), an important spe- omous systems.’’ Because of the complexity of nonlinear cial case of (20), with a continuously differentiable f: D- dynamical systems, they often have to be studied class by Rn, the following criterion of stability, called the first (or class, or even case by case. It has been widely experienced indirect) method of Lyapunov, is very convenient to use. that the first method of Lyapunov does work for some, perhaps not too many, specific nonautonomous systems in Theorem 1 (First Method of Lyapunov: For Continuous-Time case studies (e.g., in the study of some chaotic systems [7]; Autonomous Systems). Let J ¼½@f=@x x ¼ x ¼ 0 be the system see also Theorem 18, below). The point is that one has to Jacobian evaluated at the zero equilibrium of system (19). be very careful when this method is applied to a particular If all the eigenvalues of J have a negative real part, then nonautonomous system; the stability conclusion must be the system is asymptotically stable about x ¼ 0. verified by some other means at the same time. Here, it is emphasized that a rigorous approach for as- First, note that this and the following Lyapunov theo- ymptotic stability analysis of general nonautonomous sys- rems apply to linear systems as well, for linear systems tems is provided by the second method of Lyapunov, for are merely a special case of nonlinear systems. When f(x) which the following set of class K functions are useful: . ¼ Ax, the linear time-invariant system x ¼ Ax has the only equilibrium x ¼ 0. If A has all eigenvalues with neg- K¼fgðtÞ : gðt0Þ¼0; gðtÞ > 0ift > t0; ative real parts, Theorem 1 implies that the system is as- ymptotically stable about its equilibrium since the system gðtÞ is continuous and nondecreasing on ½t0; 1Þg Jacobian is simply J ¼ A. This is consistent with the familiar linear stability results. Theorem 2 (Second Method of Lyapunov: For Continuous- Note also that the region of asymptotic stability given Time Nonautonomous Systems). The system (20) is globally by Theorem 1 is local, which can be quite large for some (over the entire domain D), uniformly (with respect to the nonlinear systems but may be very small for some others. initial time over the entire time interval [t0, N)), and However, there is no general criterion for determining the asymptotically stable about its zero equilibrium, if there STABILITY OF NONLINEAR SYSTEMS 4887 exist a scalar-valued function, V(x, t), defined on D has a unique positive definite and symmetric matrix T [t0, N), and three functions a( . ), b( . ), g( . )AK, such that solution, P. Using the Lyapunov function V(x, t) ¼ x Px, it can be easily verified that
(a) V(0, t0) ¼ 0. . T T T (b) V(x, t)40 for all xa0inD and all t t0. Vðx; tÞ¼x ½PA þ A P x þ 2x Pgðx; tÞ (c) a.(||x||) V(x, t) b(||x||) for all t t0. xTx þ 2l ðPÞcjjxjj2 (d) Vðx; tÞ gðjjxjjÞo0 for all t t0. max
where lmax(P) is the largest eigenvalue of P. Therefore, if In Theorem 2, the function V is called a Lyapunov func- the constant c o 1/(2lmax(P)) and if the class K functions tion. The method of constructing a Lyapunov function for stability determination is called the second (or direct) aðzÞ¼l ðPÞz2; bðzÞ¼l ðPÞz2; method of Lyapunov. min max The geometric meaning of a Lyapunov function used for 2 gðzÞ¼½1 2clmaxðPÞ z determining the system stability about the zero equilibri- um may be illustrated by Fig. 9. In this figure, assuming are used, then conditions (c) and (d) of Theorem 2 are sat- that a Lyapunov function, V(x), has been found, which has isfied. As a result, the above system is globally, uniformly, a bowl shape as shown based on conditions (a) and (b). and asymptotically stable about its zero equilibrium. This Then, condition (d) is example shows that the linear part of a weakly nonlinear nonautonomous system can indeed dominate the stability. . @V . VðxÞ¼ xo0 ð21Þ Note that in Theorem 2, the uniform stability is guar- @x anteed by the class K functions a, b, g stated in conditions (c) and (d), which is necessary since the solution of a non- where [@V/@x] is the gradient of V along the trajectory x.It autonomous system may sensitively depend on the initial is known, from calculus, that if the inner product of this time, as seen from the numerical example discussed in . gradient and the tangent vector x is constantly negative, Section 3.1. For autonomous systems, these class K func- as guaranteed by condition (21), then the angle between tions [hence, condition (c)] are not needed. In this case, these two vectors is larger than 901, so that the surface of Theorem 2 reduces to the following simple form. V(x) is monotonically decreasing to zero (this is visualized in Fig. 9). Consequently, the system trajectory x, the pro- Theorem 3 (Second Method of Lyapunov: For Continuous- jection on the domain as shown in the figure, converges to Time Autonomous Systems). The autonomous system (19) zero as time evolves. is globally (over the entire domain D) and asymptotically As an example, consider the following nonautonomous stable about its zero equilibrium, if there exists a scalar- system valued function V(x), defined on D, such that
. x ¼ Ax þ gðx; tÞ (a) V(0) ¼ 0. a (b) V. (x)40 for all x 0inD. (c) VðxÞ 0 for all xa0inD. where A is a stable constant matrix and g is a nonlinear o function satisfying g(0, t) ¼ 0 and ||g(x, t)|| c||x|| for Note that if condition (c) in Theorem 3 is replaced by (d) a constant c40 for all tA[t , N). Since A is stable, the fol- . 0 VðxÞ 0 for all xAD, then the resulting stability is only in lowing Lyapunov equation the sense of Lyapunov but may not be asymptotic. For ex- ample, consider a simple model of an undamped pendulum T PA þ A P þ I ¼ 0 of length ‘ described by 8 < . g x ¼ sinðyÞ ‘ V(x) : . y ¼ x
where x ¼ y is the angular variable defined on po yo p, with the vertical axis as its reference, and g is the gravity constant. Since the system Jacobian at the zero equilibri- um has one pair of purely imaginary eigenvalues, pffiffiffiffiffiffiffiffiffiffiffi V l ¼ g=‘, Theorem 1 is not applicable. However, if x 1;2 one uses the Lyapunov function x 0 g 1 V ¼ ð1 cosðyÞÞ þ x2 ‘ 2 . x . then it can be easily verified that V ¼ 0 over the entire Figure 9. Geometric meaning of the Lyapunov function. domain. Thus, the conclusion is that the undamped 4888 STABILITY OF NONLINEAR SYSTEMS pendulum is stable in the sense of Lyapunov but may not As a special case, for discrete-time ‘‘autonomous’’ sys- be asymptotically, consistent with the physics of the un- tems, Theorem 6 reduces to the following simple form. damped pendulum. Theorem 7 (Second Method of Lyapunov: For Discrete-Time Theorem 4 (Krasovskii Theorem: For Continuous-Time ‘‘Autonomous’’ Systems). Let x ¼ 0 be an equilibrium for Autonomous Systems). For the autonomous system (19), the ‘‘autonomous’’ system (22). Then the system is globally let J(x) ¼ [qf/qx] be its Jacobian evaluated at x(t). A suf- (over the entire domain D) and asymptotically stable ficient condition for the system to be asymptotically stable about this zero equilibrium if there exists a scalar-valued about its zero equilibrium is that there exist two real pos- function, V(xk), defined on D and continuous in xk, such itive definite and symmetric constant matrices, P and Q, that such that the matrix (a) V(0) ¼ 0. T J ðxÞP þ PJðxÞþQ (b) V(xk)40 for all xka0inD. (c) DV(xk): ¼ V(xk) V(xk 1)o 0 for all xka0inD. is seminegative definite for all xa0 in a neighborhood D (d) V(x)-N as ||x||-N. of the origin. For this case, a Lyapunov function is given by To this end, it is important to emphasize that all the VðxÞ¼fTðxÞPfðxÞ Lyapunov theorems stated above only offer sufficient con- ditions for asymptotic stability. On the other hand, usually Furthermore, if D¼Rn and V(x)-N as ||x||-N, then more than one Lyapunov function may be constructed for this asymptotic stability is also global. the same system. For a given system, one choice of a Lyapunov function may yield a less conservative result Similar stability criteria can be established for discrete- (e.g., with a larger stability region) than other choices. time systems [10]. Two main results are summarized as However, no conclusion regarding stability may be drawn follows. if, for technical reasons, a satisfactory Lyapunov function cannot be found. Nevertheless, there is a necessary con- Theorem 5 (First Method of Lyapunov: For Discrete-Time dition in theory about the existence of a Lyapunov func- ‘‘Autonomous’’ Systems). Let x ¼ 0 be an equilibrium of tion [15], as follows. the discrete-time ‘‘autonomous’’ system Theorem 8 (Massera Inverse Theorem). Suppose that the autonomous system (19) is asymptotically stable about its xk þ 1 ¼ fðxkÞð22Þ equilibrium x and f is continuously differentiable with n A N where f : D-R is continuously differentiable in a neigh- respect to x for all t [t0, ). Then a Lyapunov function borhood of the origin, DDRn, and let J ¼½@f=@x exists for this system. k xk ¼ x ¼0 be the Jacobian of the system evaluated at this equilibri- um. If all the eigenvalues of J are strictly less than one in absolute value, then the system is asymptotically stable 4.2. Some Instability Theorems about its zero equilibrium. Once again, consider a general autonomous system Theorem 6 (Second Method of Lyapunov: For Discrete-Time . x ¼ fðxÞ; xðt Þ¼x 2 Rn ð24Þ ‘‘Nonautonomous’’ Systems). Let x ¼ 0 be an equilibrium 0 0 of the ‘‘nonautonomous’’ system with an equilibrium x ¼ 0. To disprove the stability, the following instability theorems may be used. xk þ 1 ¼ fkðxkÞð23Þ
n Theorem 9 (A Linear Instability Theorem). In system (24), where fk : D!R is continuously differentiable in a DD n let J ¼½@f=@x x ¼ x ¼ 0 be the system Jacobian evaluated at neighborhood of the origin, R . Then, system (22) is D x ¼ 0. If at lease one of the eigenvalues of J has a positive globally (over the entire domain ) and asymptotically stable about its zero equilibrium, if there exists a scalar- real part, then x ¼ 0 is unstable. For discrete-time systems, there is a similar result. A valued function, V(xk, k), defined on D and continuous in discrete-time ‘‘autonomous’’ system xk, such that
x ¼ fðx Þ; k ¼ 0; 1; 2; ; (a) V(0, k) ¼ 0 for all k k0. k þ 1 k (b) V(xk, k)40 for all xka0inD and for all k k0. (c) DV(xk, k):¼ V(xk, k) V(xk 1, k 1) o 0 for all is unstable about its equilibrium x ¼ 0 if at least one of xka0inD and all k k0 þ 1. the eigenvalues of the system Jacobian is larger than 1 in (d) 0o W(||xk||)o V(xk, k) for all k k0 þ 1, where absolute value. W(t) is a positive continuous function defined on D, N satisfying Wðjjxk0 jjÞ ¼ 0 and limt-N W(t) ¼ The following two negative theorems can be easily ex- monotonically. tended to nonautonomous systems in an obvious way. STABILITY OF NONLINEAR SYSTEMS 4889
Theorem 10 (A General Instability Theorem). For system with the Lyapunov function (24), let V(x) be a positive and continuously differentiable function defined on a neighborhood D of the origin, satis- V ¼ x2 y4 fying V(0) ¼ 0. Assume that in any subset, containing the origin, of D, there is an x~ such that V(x~ )40. If, moreover which is positive inside the region defined by d VðxÞ > 0 for all xO0inD x ¼ y2 and x ¼ y2 dt then the system is unstable about the equilibrium x ¼ 0. Let D be the right half-plane and O be the shaded area shown in Fig.. 11. Clearly, V ¼ 0 on the boundary of O, and 3 One example is the system V40 and V ¼ 2x > 0 for all (x, y)A D. According to the Chetaev theorem, this system is unstable about its zero ( . x ¼ y þ xðx2 þ y4Þ equilibrium. . y ¼ x þ yðx2 þ y4Þ 4.3. LaSalle Invariance Principle which has equilibrium (x , y ) ¼ (0, 0). The system Jacobi- Consider again the autonomous system (24) with an equilibrium x ¼ 0. Let V(x) be a Lyapunov function an at thep equilibriumffiffiffiffiffiffiffi has a pair of imaginary eigenvalues, l1;2 ¼ 1, so Theorem 1 is not applicable. On the con- defined on a neighborhood D of the origin. Let also jt(x0) trary, the Lyapunov function be a bounded solution orbit of the system, with the initial state x0 and all its limit states being confined in D.More- 1 over, let V ¼ ðx2 þ y2Þ 2 . . E ¼fx 2DjVðxÞ¼0gð25Þ leads to V ¼ðx2 þ y2Þðx2 þ y4Þ > 0 for all (x, y)a(0, 0). There- fore, the conclusion is that this system is unstable about and M E be the largest invariant subset of E in the sense its zero equilibrium. that if the initial state x0AM, then the entire orbit jt(x0) M for all t t0. Theorem 11 (Chetaev Instability Theorem). For system (24), let V(x) be a positive and continuously differentiable func- Theorem 12 (LaSalle Invariance Principle). Under the as- tion defined on D, and let O be a subset, containing the . sumptions above, for any initial state x AD, the solution origin, of D, (i.e., 0AD O). If (a) V(x)40 and VðxÞ > 0 for 0 - orbit satisfies j ðx Þ!M as t !1. all xa0inD and (b) V(x) ¼ 0 for all x on the boundary t 0 This invariance principle is consistent with the Lyapu- of O, then the system is unstable about the equilibrium nov theorems when they are applicable to a problem x ¼ 0. . [11,19]. Sometimes, when V ¼ 0 over a subset of the domain of V, a Lyapunov theorem is not easy to directly This instability theorem is illustrated by Fig. 10, which apply, but the above LaSalle invariance principle may be graphically shows that if the theorem conditions are sat- convenient to use. For instance, consider the system isfied, then there is a gap within any neighborhood of the origin, so that a system trajectory can escape from the 8 < . 1 neighborhood of the origin along a path in this gap [23]. x ¼ x þ x3 þ y 3 As an example, consider the system : . y ¼ x ( . x ¼ x2 þ 2y5 2 2 . The Lyapunov function V ¼ x þ y yields y ¼ xy2 . 1 1 V ¼ x2 x2 1 . 2 3 V (x) > 0
Ω y ) = 0 x ( V D Ω
) = 0 x V (x 0 D 0
Figure 10. Illustration of the Chetaev theorem. Figure 11. The defining region of a Lyapunov function. 4890 STABILITY OF NONLINEAR SYSTEMS
2 2 T which is negative for x o 3butiszeroforx ¼ 0andx ¼ 3, nonautonomous system (26), vðx; tÞ¼½V1ðx; tÞ Vnðx; tÞ regardless of variable y.Thus,Lyapunovtheoremsdonot in which each Vi is a continuous Lyapunov function for the seem to be applicable, at least not directly. However, observe system, i ¼ 1; ...; n; satisfying ||v(x, t)||40 for xa0. that thepffiffiffi set E defined abovepffiffiffi has only three straight lines: Assume that x ¼ 3, x ¼ 0, and x ¼ 3, and that all trajectories that . intersect the line x ¼ 0willremainonthelineonlyify ¼ 0. vðx; tÞ gðvðx; tÞ; tÞ componentwise This means that the largest invariant subset M containing the points with x ¼ 0 is the only point (0,0). It then follows for a continuous and quasimonotonic function g defined on from the LaSalle invariance principle that starting from any D. Then initial state located in a neighborhoodpffiffiffi of the origin bounded within the two stripes x ¼ 3,say,locatedinsidethedisk (a) If the system . D¼fðx; yÞjx2 þ y2o3g yðtÞ¼gðy; tÞ the solution orbit will always be attracted to the point (0, 0). is stable in the sense of Lyapunov (or asymptotical- This means that the system is (locally) asymptotically stable ly stable) about its zero equilibrium y* ¼ 0, then so about its zero equilibrium. is the nonautonomous system (26). (b) If, moreover, ||v(x, t)|| is monotonically decreas- ing with respect to t and the preceding stability (or 4.4. Comparison Principle and Vector Lyapunov Functions asymptotic stability) is uniform, then so is the non- For large-scale and interconnected nonlinear (control) sys- autonomous system (26). tems, or systems described by differential inequalities (c) If, furthermore, ||v(x, t)|| c||x||s for two pos- rather than differential equations, the stability criteria itive constants c and s, and the proceeding stability above may not be directly applicable. In such cases, (or asymptotic stability) is exponential, then so is the comparison principle and vector Lyapunov function the nonautonomous system (26). methods turn out to be advantageous [18,24,32]. To introduce the comparison principle, consider the A simple and frequently used comparison function is general nonautonomous system . jjhðy; tÞjj ¼ ð Þ ð Þ¼ ð Þ gðy; tÞ¼Ay þ hðy; tÞ; lim ¼ 0 x f x; t ; x t0 x0 26 jjyjj!0 jjyjj where f(0, t) ¼ 0 is continuous on a neighborhood D of the where A is a stable M matrix (Metzler matrix). Here, origin, t0 to1. In this case, since f is only continuous (but not neces- A ¼ [aij]isanM matrix if sarily satisfying the Lipschitz condition), this differential O equation may have more than one solution [12]. Let aiio0 and aii 0 ði jÞ; i; j ¼ 1; ...; n xmax(t) and xmin(t) be its maximum and minimum solu- tions, respectively, in the sense that 4.5. Orbital and Structural Stability Theorems Theorem 15 (Orbital Stability Theorem). Let x~ ðtÞ be a p-pe- x ðtÞ xðtÞ x ðtÞ componentwise; for all t 2½t ; 1Þ min max 0 riodic solution of an autonomous system. Suppose that the system has Floquet multipliers li, with l1 ¼ 0 and |li| 1 where x(t) is any solution of the equation, and xmin(t0) ¼ o for i ¼ 2; ...; n. Then this periodic solution x~ ðtÞ is orbitally x(t0) ¼ xmax(t0) ¼ x0. stable. Theorem 13 (The Comparison Principle). Let y (t)bea solution of the following differential inequality: Theorem 16 (Peixoto Structural Stability Theorem). Consid- er a two-dimensional autonomous system. Suppose that f . is twice differentiable on a compact and connected subset yðtÞ fðy; tÞ with yðt0Þ x0 componentwise D bounded by a simple closed curve G, with an outward If xmax(t) is the maximum solution of system (26), then normal vector n~. Assume that f n~O0onG. Then the sys- tem is structurally stable on D if and only if yðtÞ xmaxðtÞ componentwise for all t 2½t0; 1Þ (a) All equilibria are hyperbolic. The next theorem is established based on this compari- (b) All periodic orbits are hyperbolic. son principle. First, recall that a vector-valued function, (c) If x and y are hyperbolic saddles (probably x ¼ y), T s u gðx; tÞ¼½g1ðx; tÞ gnðx; tÞ is said to be quasimonotonic,if then W (x) and W (y) are transversal.
xi ¼ x~i and xj x~j ðjOiÞ 5. LINEAR STABILITY OF NONLINEAR SYSTEMS ) giðx; tÞ giðx~ ; tÞ; i ¼ 1; ; n The first method of Lyapunov provides a linear stability Theorem 14 (Vector Lyapunov Function Theorem). Let analysis for nonlinear autonomous systems. In this sec- v (x, t) be a vector Lyapunov function associated with the tion, the following general nonautonomous system is STABILITY OF NONLINEAR SYSTEMS 4891 considered: where Q is a positive definite and symmetric constant matrix. If . n x ¼ fðx; tÞ; xðt0Þ¼x0 2 R ð27Þ jjgðx; tÞjj ajjxjj which is assumed to have an equilibrium x ¼ 0. 1 N for a constant ao2 lmaxðPÞ uniformly on [t0, ), where 5.1. Linear Stability of Nonautonomous Systems lmax(P) is the maximum eigenvalue of P, then system (28) is globally, uniformly, and asymptotically stable about its In system (27), Taylor-expanding the function f about equilibrium x ¼ 0. x ¼ 0 gives . x ¼ fðx; tÞ¼JðtÞx þ gðx; tÞð28Þ This actually is the example for illustration of Theo- rem 2 discussed in Section 4.1. Again consider system (27) with equilibrium x ¼ 0. where JðtÞ¼½@f=@x x ¼ 0 is the Jacobian and g(x, t) is the residual of the expansion, which is assumed to satisfy Theorem 19 (The Lyapunov Converse Theorem). Suppose 2 that f is continuously differentiable in a neighborhood of jjgðx; tÞjj ajjxjj for all t 2½t0; 1Þ the origin, and its Jacobian J(t) is bounded and Lipschitz in the neighborhood, uniformly in t. Then, the system is where a40 is a constant. It is known from the theory of exponentially stable about its equilibrium if and only if its elementary ordinary differential equations [12] that the . linearized system x ¼ JðtÞx is exponentially stable about solution of equation (28) is given by the origin. Z t xðtÞ¼Fðt; t0Þx0 þ Fðt; tÞgðxðtÞ; tÞdt ð29Þ 5.2. Linear Stability of Nonlinear Systems with t 0 Periodic Linearity where F(t, t) is the fundamental matrix associated with Consider a nonlinear nonautonomous system of the form matrix J(t). . n x ¼ fðx; tÞ¼JðtÞx þ gðx; tÞ; xðt0Þ¼x0 2 R ð30Þ Theorem 17 (A General Linear Stability Theorem). For the nonlinear nonautonomous system (28), if there are two where g(0, t) ¼ 0 and J(t)isap-periodic matrix (p40): positive constants c and s such that
Jðt þ pÞ¼JðtÞ for all t 2½t0; 1Þ sðt tÞ jjFðt; tÞjj ce for all t0 t to1 Theorem 20 (Floquet Theorem). In system (30), assume and if that g(x, t) and qg(x, t)/qx are both continuous in a bound- ed region D containing the origin. Assume, moreover, that jjgðx; tÞjj lim ¼ 0 jjxjj!0 jjxjj jjgðx; tÞjj lim ¼ 0 jjxjj!0 jjxjj uniformly with respect to tA[t0, N), then there are two positive constants, g and d, such that uniformly on [t0, N). If the system Floquet multipliers satisfy gðt t0Þ jjxðtÞjj cjjx0jje
jlijo1; i ¼ 1; ; n; for all t 2½t0; 1Þ ð31Þ for all jjx0jj d and all tA[t0, N). then system (30) is globally, uniformly, and asymptotically This result implies that under the theorem conditions, stable about its equilibrium x ¼ 0. the system is locally, uniformly, and exponentially stable about its equilibrium x ¼ 0. Note that if g(x, t) ¼ 0 in system (28) or (30), all the In particular, if the system matrix J(t) ¼ J is a stable linear stability results presented above still hold and are constant matrix, then the following simple criterion is consistent with the familiar results from the linear sys- convenient to use. tems theory.
Theorem 18 (A Special Linear Stability Theorem). Suppose that in system (28), the matrix J(t) ¼ J is a stable constant 6. TOTAL STABILITY: STABILITY UNDER matrix (all its eigenvalues have a negative real part), and PERSISTENT PERTURBANCES g(0, t) ¼ 0. Let P be a positive definite and symmetric matrix solution of the Lyapunov equation Consider a nonautonomous system of the form
T . n PJ þ J P þ Q ¼ 0 x ¼ fðx; tÞþhðx; tÞ; xðt0Þ¼x0 2 R ð32Þ 4892 STABILITY OF NONLINEAR SYSTEMS where f is continuously differentiable, with f(0, t) ¼ 0, and This can be implemented via the block diagram shown in h is a persistent perturbation in the sense that for any Fig. 12, where, for notational convenience, both time- and e > 0, there are two positive constants, d1 and d2, such that frequency-domain symbols are mixed. if jjhðx~ ; tÞjjod1 for all tA[t0, N) and if jjx~ ðt0Þjjod2, then The Lur’e system shown in Fig. 12 is a closed-loop con- jjx~ ðtÞjjoe. figuration, where the feedback loop is usually considered The equilibrium x ¼ 0 of the unperturbed system [sys- as a ‘‘controller.’’ Thus, this system is sometimes written in tem (32) with h ¼ 0 therein] is said to be totally stable,if the following equivalent form: the persistently perturbed system (32) remains to be sta- 8 ble in the sense of Lyapunov. > . <> x þ Ax þ Bu As the next theorem states, all uniformly and asymp- > y ¼ Cx ð38Þ totically stable systems with persistent perturbations are :> totally stable, namely, a stable orbit starting from a neigh- u ¼ hðyÞ borhood of another orbit will stay nearby [13,15].
Theorem 21 (Malkin Theorem). If the unperturbed system 7.1. SISO Lur’e Systems (32), that is, with h ¼ 0 therein, is uniformly and asymp- First, single-input/single-output Lur’e systems are dis- totically stable about its equilibrium x ¼ 0, then it is to- cussed, where u ¼ h(y) and y ¼ cTx are both scalar-valued tally stable, namely, the persistently perturbed system functions: (32) remains to be stable in the sense of Lyapunov. 8 . > x ¼ Ax þ bu Next, consider an autonomous system with persistent <> T perturbations: > y ¼ c x ð39Þ :> . u ¼ hðyÞ x ¼ fðxÞþhðx; tÞ; x 2 Rn ð33Þ Assume that h(0) ¼ 0, so that x ¼ 0 is an equilibrium of the system.
If j (x ) Theorem 22 (Perturbed Orbital Stability Theorem). t 0 7.1.1. The Sector Condition. The Lur’e system (39) is is an orbitally stable solution of the unperturbed autono- said to satisfy the local (global) sector condition on the mous system (33) (with h ¼ 0 therein), then it is totally nonlinear function h( ), if there exist two constants, ao b, stable, that is, the perturbed system remains to be orbi- such that tally stable under persistent perturbations. 1. Local sector condition:
2 2 7. ABSOLUTE STABILITY AND FREQUENCY- ay ðtÞ yðtÞhðyðtÞÞ by ðtÞ for all ð40Þ DOMAIN CRITERIA a yðtÞ b and t 2½t0; 1Þ Consider a feedback system in the Lur’e form 2. Global sector condition: ( . x ¼ Ax þ BhðyÞ ð34Þ ay2ðtÞ yðtÞhðyðtÞÞ by2ðtÞ for all y ¼ Cx ð41Þ 1oyðtÞo1 and t 2½t0; 1Þ where A, B, C are constant matrices, in which A is non- singular but B and C are not necessarily square (yet, Here, [a, b] is called the sector for the nonlinear function probably, B ¼ C ¼ I), and h is a vector-valued nonlinear h( ). Moreover, the system (39) is said to be (globally) function. By taking the Laplace transform with zero ini- absolutely stable within the sector [a, b] if the system tial conditions, and denoting the transform by x^ ¼Lfxg, is (globally) asymptotically stable about its equilibrium the state vector is obtained as x ¼ 0 for any nonlinear function h( ) satisfying the
x^ ¼½sI A 1BLfhðyÞg ð35Þ so that the output is given by + x y G (s) C − y^ ¼ CGðsÞLfhðyÞg ð36Þ with the system transfer matrix h (y)
1 GðsÞ¼½sI A B ð37Þ Figure 12. Configuration of the Lur’e system. STABILITY OF NONLINEAR SYSTEMS 4893
y(t) y(t) [17,26,27,29]. A more direct generalization of the Nyquist h(y(t)) criterion to nonlinear systems is the following. y(t) y(t) h(y(t)) Theorem 24 (Circle Criterion). Suppose that the SISO a Lur’e system (39) satisfies the following conditions: 0 b y (t) 0 y(t) (a) A has no purely imaginary eigenvalues, and has k (a) (b) eigenvalues with positive real parts. (b) The system satisfies the global sector condition. Figure 13. Local and global sector conditions. (c) One of the following situation holds: (1) 0o ao b—the Nyquist plot of G(jo) encircles the (global) sector condition. These local and global sector con- disk D[ (1/a), (1/b)] counterclockwise k times ditions are visualized by Figs. 13a and 13b, respectively. but does not enter it. (2) 0 ¼ ao b—the Nyquist plot of G(jo) stays within 7.1.2. The Popov Criterion. the open half-plane Re{s}4 (1/b). (3) ao 0o b—the Nyquist plot of G(jo) stays within Theorem 23 (Popov Criterion). Suppose that the SISO the open disk D[ (1/b), (1/a)]. Lure’s system (39) satisfied the following conditions: (4) aobo0—the Nyquist plot of G(jo) encircles the disk D[(1/a), (1/b)] counterclockwise k times (a) A is stable and {A, b} is controllable. but does not enter it. (b) The system satisfies the global sector condition Then, the system is globally asymptotically stable about with a ¼ 0 therein. its equilibrium x ¼ 0. (c) For any e > 0, there is a constant g40 such that Here, the disk D[ (1/a), (1/b)], for the case of 1 Refð1 þ jgoÞGðjoÞg þ e for all o 0 ð42Þ 0oaob, is shown in Fig. 15. b 7.2. MIMO Lur’e systems where G(s) is the transfer function defined by (37), Consider a multiinput/multioutput Lur’e system, as and Re{ } denotes the real part of a complex num- shown in Fig. 12, described by ber (or function). Then, the system is globally as- ymptotically stable about its equilibrium x ¼ 0 ( xðsÞ¼GðsÞuðsÞ within the sector. ð43Þ uðtÞ¼ hðyðtÞÞ The Popov criterion has the following geometric mean- ing. Separate the complex function G(s) into its real and with G(s) as defined in (37). If this system satisfies the imaginary parts, namely following Popov inequality Z t1 GðjoÞ¼GrðoÞþjGiðoÞ T y ðtÞxðtÞdt g for all t1 t0 ð44Þ t0 and rewrite condition (C) as for a constant g 0 independent of t, then it is said to be 1 hyperstable. > G ðoÞþgoG ðoÞ for all o 0 b r i The linear part of this MIMO system is described by the transfer matrix G(s), which is said to be positive real if Then the graphical situation of the Popov criterion shown in Fig. 14 implies the global asymptotic stability of the 1. There are no poles of G(s) located inside the open system about its zero equilibrium. half-plane Re{s}40. 2. Poles of G(s) on the imaginary axis are simple, and 7.1.3. The Circle Criterion. The Popov criterion has a the residues form a semipositive definite matrix. natural connection to the linear Nyquist criterion
1 − Gr( ) + Gi ( )
1 1 0 0
Figure 14. Geometric meaning of the Popov criterion. Figure 15. The disk D[ (1/a), (1/b)]. 4894 STABILITY OF NONLINEAR SYSTEMS
3. The matrix [G(jo) þ GT(jo)] is a semipositive defi- 1. If the two curves are (almost) tangent, as illustrated nite matrix for all real values of o that are not poles by Fig. 16a, then a conclusion drawn from the de- of G(s). scribing function method will not be satisfactory in general. 2. If the two curves are (almost) transversal, as illus- Theorem 25 (Hyperstability Theorem). The MIMO Lur’e trated by Fig. 16b, then a conclusion drawn from the system (43) is hyperstable if and only if its transfer describing function analysis will generally be reliable. matrix G(s) is positive real.
Theorem 27 (Graphical Stability Criterion for a Periodic 7.3. Describing Function Method Orbit). Each intersection point of the two curves above, h1i Return to the SISO Lur’e system (39) and consider its pe- Gr(jo) and 1/C(a), corresponds to a periodic orbit, y ðtÞ, riodic output y(t). Assume that the nonlinear function h( ) of the output of system (39). If the points, near the inter- therein is a time-invariant odd function, and satisfies the section and on one side of the curve 1/C(a) where a is property that for y(t) ¼ a sin(ot), with real constants o and increasing, are not encircled by the curve Gr(jo), then the aa0, only the first-order harmonic of h(y) in its Fourier corresponding periodic output is stable; otherwise, it is series expansion is significant. Under this setup, the spe- unstable. cially defined function Z 2o p=o CðaÞ¼ hða sinðotÞÞ sinðotÞ dt ð45Þ 8. BIBO STABILITY ap 0 A relatively simple, and also relatively weak, notion of is called the describing function of the nonlinearity h( ), stability is discussed in this section. This is the bounded- or of the system [1,26,27]. input/bounded-output (BIBO) stability, which refers to the property of a system that any bounded input to the system produces a bounded output throughout the system [8,9,17]. Theorem 26 (First-Order Harmonic Balance Approxima- . tion). Under the conditions stated above, if the first-order It can be verified that the linear system x ¼ Ax þ Bu is harmonic balance equations BIBO stable if the matrix A is asymptotically stable. Here, the focus is on the input-output map (13), with its configuration shown in Fig. 2. GrðjoÞCðaÞ¼1 and Gið joÞ¼0 The system S is said to be BIBO stable from the input set U to the output set Y, if for each admissible input uAU have solutions o and aa0, then and the corresponding output yAY, there exist two non- negative constants, bi and bo, such that ja ja yh1iðtÞ¼ e jot ejot 2 2 jjujjU bi )jjyjjY bo ð46Þ is the first-order approximation of a possible periodic orbit of the output of system (39); but if the preceding harmonic Since all norms are equivalent for a finite-dimensional balance equations have no solution, then the system is vector, it is generally insignificant to distinguish under unlikely to have any periodic output. what kind of norms for the input and output signals the BIBO stability is defined and achieved. Moreover, it is im- When solving the equation Gr( jo)C(a) ¼ 1 graphically, portant to note that in the definition above, even if bi is one can sketch two curves in the complex plane: Gr( jo) small and bo is large, the system is still considered to be and 1/C(a) by increasing gradually o and a, respec- BIBO-stable. Therefore, this stability may not be very tively, to find their crossing points: practical for some systems in certain applications.
lm lm
Re Re 0 0
−1/Ψ() −1/Ψ() G(j) G(j) Figure 16. Graphical depiction of describing func- tion analysis. (a)(b) STABILITY OF NONLINEAR SYSTEMS 4895
8.1. Small-Gain Theorem Theorem 29 (Passivity Stability Theorem). If there exist four constants, L , L , M , M , with L þ L 40, such that A convenient criterion for verifying the BIBO stability of a 1 2 1 2 1 2 ( closed-loop control system is the small-gain theorem 2 he1; S1ðe1Þi L1jje1jj þ M1 [8,9,17], which applies to almost all kinds of systems 2 ð50Þ 2 (linear and nonlinear, continuous-time and discrete-time, he2; S2ðe2Þi L2jjS2ðe2Þjj2 þ M2 time-delayed, of any dimensions), as long as the mathe- matical setup is appropriately formulated to meet the the- then the closed-loop system (47) is BIBO-stable. orem conditions. The main disadvantage of this criterion is its overconservativity. As mentioned, the main disadvantage of this criterion Return to the typical closed-loop system shown in is its overconservativity in providing the sufficient condi- Fig. 3, where the inputs, outputs, and internal signals tions for the BIBO stability. One resolution is to transform are related via the following equations: the system into the Lur’e structure, and then apply the ( circle or Popov criterion under the sector condition (if it S1ðe1Þ¼e2 u2 can be satisfied), which can usually lead to less conserva- ð47Þ tive stability conditions. S2ðe2Þ¼u1 e1 8.2. Contraction Mapping Theorem It is important to note that the individual BIBO stabil- The small-gain theorem discussed above, by nature, is a ity of S and S is not sufficient for the BIBO stability of 1 2 kind of contraction mapping theorem. The contraction the connected closed-loop system. For instance, in the dis- mapping theorem can be used to determine the BIBO sta- crete-time setting of Fig. 3, suppose that S 1 and 1 bility property of a system described by a map in various S 1, with u (k) 1 for all k ¼ 0; 1; . Then S and S 2 1 1 2 forms, provided that the system (or the map) is appropri- are BIBO-stable individually, but it can be easily verified ately formulated. The following is a typical (global) con- that y (k) ¼ k-N as the discrete-time variable k evolves. 1 traction mapping theorem. Therefore, a stronger condition describing the interaction of S and S is necessary. 1 2 Theorem 30 (Contraction Mapping Theorem). If the opera- tor norm of the input–output map S, defined on Rn, sati- Theorem 28 (Small-Gain Theorem). If there exist four con- sfies |||S||| o 1, then the system mapping stants, L1, L2, M1, M2, with L1L2 o 1, such that yðtÞ¼SðyðtÞÞ þ c ( jjS1ðe1Þjj M1 þ L1jje1jj ð48Þ has a unique solution for any constant vector c 2 Rn. This jjS2ðe2Þjj M2 þ L2jje2jj solution satisfies then y ð1 jjjSjjjÞ 1kkc ( 1 In particular, the solution of the equation jje1jj ð1 L1L2Þ ðjju1jj þ L2jju2jj þ M2 þ L2M1Þ ð49Þ jje jj ð1 L L Þ 1ðjju jj þ L jju jj þ M þ L M Þ n 2 1 2 2 1 1 1 1 2 yk þ 1 ¼ SðykÞ; y0 2 R ; k ¼ 0; 1; ... where the norms || || are defined over the spaces that satisfies the signals belong. Consequently, (48) and (49) together imply that if the system inputs (u1 and u2) are bounded jjykjj ! 0ask !1 then the corresponding outputs [S1(e1) and S2(e2)] are bounded. 9. CONCLUDING REMARKS Note that the four constants, L1, L2, M1, M2, can be somewhat arbitrary (e.g., either L1 or L2 can be large) This article has offered a brief introduction to the basic provided that L1L2o 1, which is the key condition for the theory and methodology of Lyapunov stability, orbital sta- 1 theorem to hold [and is used to obtain (1 L1L2) in the bility, structural stability, and input–output stability for bounds (49)]. nonlinear dynamical systems. More subtle details for sta- In the special case where the input–output spaces U bility analysis of general dynamical systems can be found and Y are both the L2 space, a similar criterion based on in the literature [3,5,9,11–13,16,17,19,20,23,26,27,29,30]. the system passivity property can be obtained [9,17]. In In cases where control was explicitly involved, stability this case, an inner product between any two vectors in the and stabilization issues have also been studied space is defined by [8,17,25,28,33]. Several important classes of nonlinear (control) sys- Z 1 tems have been omitted in the discussion of various sta- T hx; Zi¼ x ðtÞZðtÞ dt bility issues: some general functional systems such as t0 4896 STANDING WAVE METERS AND NETWORK ANALYZERS systems with time delays [14], measure ordinary differen- 21. A. M. Lyapunov, The General Problem of Stability of Motion tial equations such as systems with impulses [2,31], and (100th Anniversary), Taylor & Francis, London, 1992. some weakly nonlinear systems such as piecewise linear 22. A. A. Martynyuk, ed., Advances in Stability Theory at the End and switching (non)linear systems. Discussion of more ad- of the 20th Century, Taylor & Francis, London, 2003. vanced nonlinear systems, such as singular nonlinear sys- 23. D. R. Merkin, Introduction to the Theory of Stability, Spring- tems (perhaps with time delays), infinite-dimensional er-Verlag, New York, 1997. (non)linear systems, spatiotemporal systems described 24. A. N. Michel and R. K. Miller, Qualitative Analysis of Large by nonlinear partial differential equations, and nonlinear Scale Dynamical Systems, Academic Press, New York, 1977. stochastic (control) systems, are beyond the scope of this 25. N. Minorsky, Theory of Nonlinear Control Systems, New York: elementary expository article. McGraw-Hill, 1969. Finally, for more recent studies of system stability 26. R. R. Mohler, Nonlinear Systems, Vol. 1, Dynamics and theories, the reader is referred to the reviews edited by Control, Prentice-Hall, Englewood Cliffs, NJ, 1991. Martynyuk [22]. 27. K. S. Narendra and J. H. Taylor, Frequency Domain Criteria for Absolute Stability, Academic Press, New York, 1973. 28. H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical BIBLIOGRAPHY Control Systems, Springer-Verlag, New York, 1990. 29. P. C. Parks and V. Hahn, Stability Theory, Prentice-Hall, 1. D. P. Atherton, Nonlinear Control Engineering (student ed.), New York, 1981. Van Nostrand Reinhold, New York, 1982. 30. S. Yu. Pilyugin, Introduction to Structurally Stable Systems 2. D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: of Differential Equations, Birkha¨ser, Boston, 1992. Stability, Theory and Applications, Ellis Horwood, Chichester 31. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential (UK), 1989. Equations, World Scientific, Singapore, 1995. 3. N. P. Bhatia and G. P. Szego¨, Stability Theory of Dynamical 32. D. D. Siljack, Large Scale Dynamic Systems: Stability and Systems, Springer-Verlag, Berlin, 1970. Structure, North-Holland, Amsterdam, 1978. 4. F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary 33. E. D. Sontag, Mathematical Control Theory: Deterministic Differential Equations: An Introduction, 1969; Dover, New Finite Dimensional Systems, 2nd ed., Springer-Verlag, York, 1989. New York, 1998. 5. L. Cesari, Asymptotic Behavior and Stability Problems in Or- 34. F. Verhulst, Nonlinear Differential Equations and Dynamical dinary Differential Equations, Springer-Verlag, New York, Systems, Springer-Verlag, Berlin, 1990. 1971. 35. M. Vidyasagar, Nonlinear Systems Analysis, 2nd ed., Pren- 6. G. Chen, G. Chen, and S. H. Hsu, Linear Stochastic Control tice-Hall, Englewood Cliffs, NJ, 1993. Systems, CRC Press, Boca Raton, FL, 1995. 36. M. Y. Wu, A note on stability of linear time-varying systems, 7. G. Chen and X. Dong, From Chaos to Order: Methodologies, IEEE Trans. Automat. Control (April 1974), p. 162. Perspectives and Applications, World Scientific, Singapore, 1998. 37. N. E. Zhukovskij, On solidity of motion, in Collection of Works, Leningrad Press, Moscow, 1948, Vol. 1, p. 67. 8. R. J. P. de Figueiredo and G. Chen, Nonlinear Feedback Con- trol Systems: An Operator Theory Approach, Academic Press, San Diego, 1993. 9. C. A. Desoer and M. Vidyasagar, Feedback Systems: Input– Output Properties, Academic Press, New York, 1975. 10. H. Freeman, Discrete-Time Systems: An Introduction to the STANDING WAVE METERS AND Theory, Wiley, New York, 1965. NETWORK ANALYZERS 11. P. Glendinning, Stability, Instability and Chaos, Cambridge Univ. Press, New York, 1994. BURKHARD SCHIEK 12. R. Grimshaw, Nonlinear Ordinary Differential Equations, ANDREAS GRONEFELD CRC Press, Boca Raton, FL, 1993. Ruhr-Universita¨t Bochum 13. W. Hahn, Stability of Motion, Springer-Verlag, Berlin, 1967. 14. J. Hale, Theory of Functional Differential Equations, Spring- er-Verlag, New York, 1973. 1. MEASUREMENT OF NETWORKS AT 15. F. C. Hoppensteadt, Analysis and Simulation of Chaotic MICROWAVE FREQUENCIES Systems, Springer-Verlag, New York, 1977. 16. D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Determining the characteristics of electrical networks Equations, 2nd ed., Oxford Univ. Press, New York, 1987. (network analysis) is an important measurement tool 17. H. K. Khalil, Nonlinear Systems, 2nd ed., Prentice-Hall, both for developers of electrical circuits and systems and Upper Saddle River, NJ, 1996. in production testing. The measurement of network char- 18. V. Lakshmikantham, V. M. Matrosov, and S. Sivasundaram, acteristics at microwave frequencies employs different Vector Lyapunov Functions and Stability Analysis of Nonlin- concepts than are used at lower frequencies, where mea- ear Systems, Kluwer Academic, Boston, 1991. suring port voltages and currents readily allows determi- 19. J. P. LaSalle, The Stability of Dynamical Systems, SIAM, nation of impedance (Z) or admittance (Y) parameters. At Philadelphia, 1976. higher frequencies, neither high-impedance voltage 20. S. Lefschetz, Differential Equations: Geometric Theory, 2nd probes nor low-impedance current measurements are ed., 1963; Dover, New York, 1977. feasible, because parasitic capacitance and inductance of